B. A. Khesin, G. Misiołek, “Shock Waves for the Burgers Equation and Curvatures of Diffeomorphism Groups”, Analysis and singularities. Part 2, Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 259, Nauka, MAIK «Nauka/Inteperiodika», M., 2007, 77–85; Proc. Steklov Inst. Math., 259 (2007), 73

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Volume 259, Pages 77–85 (Mi tm570)

This article is cited in 11 scientific papers (total in 11 papers)

Shock Waves for the Burgers Equation and Curvatures of Diffeomorphism Groups

B. A. Khesin^a, G. Misiołek^b

^a Department of Mathematics, University of Toronto
^b Department of Mathematics, University of Notre Dame

Full-text PDF (209 kB) Citations (11)

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Abstract: We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler hydrodynamics (via Arnold's approach) to shock formation in the multidimensional Burgers equation and the Kantorovich–Wasserstein geometry of the space of densities.

Received in February 2007

English version:
Proceedings of the Steklov Institute of Mathematics, 2007, Volume 259, Pages 73–81
DOI: https://doi.org/10.1134/S0081543807040062

Bibliographic databases:

Language: English

Citation: B. A. Khesin, G. Misiołek, “Shock Waves for the Burgers Equation and Curvatures of Diffeomorphism Groups”, Analysis and singularities. Part 2, Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 259, Nauka, MAIK «Nauka/Inteperiodika», M., 2007, 77–85; Proc. Steklov Inst. Math., 259 (2007), 73–81

Citation in format AMSBIB

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https://www.mathnet.ru/eng/tm570

https://www.mathnet.ru/eng/tm/v259/p77

This publication is cited in the following 11 articles:

Boris Khesin, Gerard Misiołek, Alexander Shnirelman, “Geometric Hydrodynamics in Open Problems”, Arch Rational Mech Anal, 247:2 (2023)
Pan K., Wu X., Yue X., Ni R., “A Spatial Sixth-Order Ccd-Tvd Method For Solving Multidimensional Coupled Burgers' Equation”, Comput. Appl. Math., 39:2 (2020), 76
Sergey V. Zakharov, “Asymptotic solutions of a parabolic equation near singular points of $A$ and $B$ types”, Ural Math. J., 5:1 (2019), 101–108
Chen B., He D., Pan K., “A Linearized High-Order Combined Compact Difference Scheme For Multi-Dimensional Coupled Burgers' Equations”, Numer. Math.-Theory Methods Appl., 11:2 (2018), 299–320
Khanin K., Sobolevski A., “On Dynamics of Lagrangian Trajectories for Hamilton–Jacobi Equations”, Arch. Ration. Mech. Anal., 219:2 (2016), 861–885
S. V. Zakharov, “Singularities of $A$ and $B$ Types in Asymptotic Analysis of Solutions of a Parabolic Equation”, Funct. Anal. Appl., 49:4 (2015), 307–310
Khesin B., Lenells J., Misiolek G., Preston S.C., “Curvatures of Sobolev Metrics on Diffeomorphism Groups”, Pure Appl. Math. Q., 9:2, 2, SI (2013), 291–332
Khanin K., Sobolevski A., “Particle dynamics inside shocks in Hamilton–Jacobi equations”, Phil. Trans. R. Soc. A, 368:1916 (2010), 1579–1593
Khesin B., Lee P., “A nonholonomic Moser theorem and optimal transport”, J. Symplectic Geom., 7:4 (2009), 381–414
Delanoe Ph., “Differential Geometric Heuristics for Riemannian Optimal Mass Transportation”, Differential Equations: Geometry, Symmetries and Integrability - the Abel Symposium 2008, Abel Symposia, 5, 2009, 49–73
Philippe Delanoë, Differential Equations - Geometry, Symmetries and Integrability, 2009, 49

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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